Electromagnetism and magnetic circuit 4

8/9/2019 Electromagnetism and magnetic circuit 4

1/15

16-Apr-10 CHETAN UPADHYAY 1

Energy In A Coupled Circuit

2Li2

1w !

Energy stored in an inductor:

21

2

22

2

11iMiiL

2

1iL

2

1w s!

Energy stored in a coupled circuit:

Positive sign: both currents enter or leave the dotted terminals

Negative sign: one current enters and one current leaves the dotted terminals

Unit : Joule


2/15


1

. .

M

2

+ +

--

1v 2v

1i

2i

Coupled Circuit



3/15


0iMiiL2

1

iL2

1

21

2

22

2

11u

Energy stored must be greater or equal to zero.

0MLL21

u21

LLM eor

Mutual inductance cannot be greater than the geometric mean of self inductances.



4/15


The coupling coefficient k is a measure of the magnetic

coupling between two coils

21LL

Mk ! 21LLkM !

1k0 ee 21LLM0 ee

or

Where:

or


1k0 ee


5/15


Perfectly coupled : k = 1

Loosely coupled : k < 0.5

- Linear/air-core transformers

Tightly coupled : k > 0.5

- Ideal/iron-core transformers

Coupling coefficient depends on :

1. The closeness of the two coils

2. Their core

3. Their orientation

4. Their winding



6/15

COMPARISON OF MAGNETIC/ELECTRIC CIRCUIT

SIMILARITIES

The closed path for magnetic flux is

called magnetic circuit.

flux = mmf / reluctance

flux in Weber

mmf in AT

reluctance = S = l/ A

permeance = 1 / reluctance

permeability

reluctivity

Flux density B = / A

Magnetic Intensity H = NI / l

The closed path for electric current

is called electric circuit.

current = emf / resistance

current I in ampere

emf in volts

resistance = R = l / A

conductance = 1 / resistance

conductivity

resistivity

current density J = I / A

Electric intensity E = V/d

16-Apr-10 6CHETAN UPADHYAY

MAGNETIC CIRCUIT ELECTRIC CIRCUIT


7/15

Comparison of electrical and magnetic circuit

Dissimilarities

Magnetic circuit

Magnetic flux does not flowbut is set up in the magnetic

circuit For magnetic flux, there is no

perfect insulator

At constant temp, thereluctance of a magneticcircuit is not constant but

varies with r Once magnetic flux is set up in

a magnetic circuit, no energy isneeded

Electrical Circuit

The electric current actuallyflows in an electric circuit.

For electric circuit, there arelarge number of perfectinsulators

At constant temp, theresistance of an electric circuitis constant as its valuedepends on resistivity which is

almost constant Energy is needed as long as

current flows through theelectric circuit.



8/15

Equivalent Inductance in parallel

The inductances can be connected in parallel

such that : -

1) the mutually induced emf assists the selfinduced emfs.

2) the mutually induced emf opposes the self

induced emfs.



9/15

CHETAN UPADHYAY 9

1L 2L

M

y y

1v

2v

1

i2i

1L 2L

M

y

y

1v

2v

1i 2i

dt

diL

dt

diMv

dt

diM

dt

diLv

2

2

1

2

2111

!

!

dt

di

dt

div

dtdi

dtdiv

22

12

2111

!

!

When the reference direction for a current enters the dotted

terminal of a coil, the reference polarity of the voltage that it

induces in the other coil is positive at its dotted terminal.

Dot convention

16-Apr-10

Suppose, both are in parallel, v1 = v2


10/15

CHETAN UPADHYAY 10

1L 2L

M

y y

1v

2v

1i

2i

Suppose, both are in parallel, v1 = v2

1L 2L

M

y

y

1v

2v

1i 2i

dt

di

Ldt

di

Mv

dt

diM

dt

diLv

2

2

1

2

2111

!

!

dt

di

Ldt

di

Mv

dt

diM

dt

diLv

2212

2111

!

!

16-Apr-10


11/15

RL circuit:

What happens to an inductor

in a circuit?



12/15

Result: Charging

/( ) (1 )t I t e

R

XI

!

= L/R= -time constant



13/15

Discharging:

/

0( )t

I t I eX

!



14/15

The switch S below is initially unconnected. At time tAthe switch is connected to point A. What is the

equation from KLR for this circuit?

1 )I Ldi

dt iR ! 0

2 ) I Ldi

dt iR ! 0

3 ) I Ldi

dt iR ! 0

4 )I Li

t

iR ! 016-Apr-10 14CHETAN UPADHYAY


15/15

Just as for an RCcircuit, the

voltage and current in an RLcircuit decay exponentially

towards their long-time

values,

(I,(V~ e-t/X

For an RCcircuit, the

time constant was

= RC

The RL Circuit

For an RL circuit, the

time constant is

X= L/R

nits: ohm-farad = sec = henry/ohm16-Apr-10 15CHETAN UPADHYAY

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Electromagnetism and magnetic circuit 4